I am planning on claiming my son on my taxes this year but I loved with my mother for part of the year and I have a feeling she is going to try to claim my child in her taxes. She thinks because her husband makes more money than me they have the right to claim my child. What happens if she does try to claim him after I already filed?

If a child is a qualifying child for 2 people, the tie breaker rule is that the parent of the child get's to claim the child.
The rule that the one with the higher income gets to claim the child would only apply if the child is the qualifying child of both parents, then the parent with the higher income would be allowed to claim the child.
If two people claim the same child, the IRS will red flag both returns and ask for supporting information and decide who gets to claim the child

Just to add - be sure to have a conversation with your mother about this. It will save a whole lot of heartache and grief if you don't both claim the child, as one of you will end up owing interest and penalties if you both try and claim him. As TfA said - you will win this dogfight, so please let your mother know that so she isn't caught by surprise.

Alternatively, if your mother is indeed in a higher tax bracket and if your son can be considered a qualifying child for her, then you may be better off working a side deal with her. That could turn it into a win-win.

Unfortunately my mother has not spoken to me in over 2 months now because I moved out of her house and she will no longer have the extra income from me and my child. That's why I have a feeling she is going to try to claim my son on her taxes this year even though I did provide for him for at least 10 months she just wants the money she could get for him.